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Velocity distribution along an elastic model of human arterial tree
VELOCITY
DISTRIBUTION ALONG AN ELASTIC OF HUMAN ARTERIAL TREE
In>tirut de hticaniqut:
des Fluidcs.
MODEL
1 Rue Honnorat. I XX73Marseille. France and
ALAIN
Unit6 Inserm 175, Cardiologie
FRIGGI and ROBERT PELISSIER ExpGmentale. lg Avenue Mozart. l?@O!J. Marseille, France
Abstract-An experimental investigation ofan elastic model ofthe human arterial tree, has ken performed ior physiological type flow by pulsed Doppler ultrasonic velocimetry. The arterial tree model, fabricated in clear poiyurrthanc, includes the aortic arch, with a Starr-Edwards ball valve mounted in the root of the aorta, the descending aorta and the iliac bifurcation. Our study showed that the velocity profile, a few centimeters beyond the valve, is skcwcd, with higher velocities towards the top and the inner wall (anatomically the posterior and left lateral wall). An inward shift of the maximum velocity and reverse flow are denoted along the inner wall of the aortic arch. The velocity profiles in the descending aorta are blunted. Downstream from the vertex ofthe iliac bifurcation. thcrc is vorticitv creation, but the branching effect is quickly damped by the pulsatility of the flow and the elasticity of the ;ail.
rit.0 attempts have been made to describe the velocity profiles in the aorta using hot-film anemometry (Ling er al., 1968; Schultz et al., 1969; Seed and Wood, 1971; Falsetti et ul., 1972; Paulsen and Hasenkam, 1983) or pulsed Doppler ultrasonic anemometry (Farthing and Peronneau, 1979). Numerous studies have been reported in the literature which investigate flow characteristics in branching channels. A large review was given by Pedley (1980). The flow velocity pattern has been considered in steady conditions by Schroter and Sudlow (1969). Olson (1971). Brech and Bellhouse (1973). Talukder (1975) and Tomm (1978). Pulsatile flow studies have been reported by Brech and Bellhouse (19731, Rieu et al. (1979). Walburn and Stein (1981), Batten and Nerem (1982) and Siouffi et a/. (1984). The rheological properties of the arterial wall are of interest in fluid dynamics. Elasticity is of importance but is neglected by most investigators in this area of research. In reality, the elastic behaviour of the wall is non-linear; this non-linearity of the wall elements was clearly proved by Ling and Atabek (1972), Pate1 et al. (1973). Taylor and Gerrard (1977), Wetterer et al. (1977) and Friggi and Bodard (1981). The purpose of the present paper was to determine, under physiological type flow conditions, the velocity profiles in several cross-sections of an elastic model of arterial tree including a Starr-Edwards valve prosthesis. Velocity measurements were performed for the longitudinal and the radial velocity components and afterwards for the velocity vectors.
INTRODUCTIOS
The description of the flow pattern in the arterial tree is of basic importance to understanding the fluid mechanics of the circulation. The very complex nature of the phenomena in the arterial tree is due to the fact that the flow is disturbed by many singularities: aortic valve, aortic arch, branches at the arch and iliac bifurcation. Valve implantations remain associated with thrombus formation and tissue overgrowth (Yoganathan et a[.. 198Oa);sites of curvature and bifurcations are known to be particularly susceptible to the development of atherosclerotic lesions (Fry, 1968; Care et al., 1971). Flow in the arterial tree has been of interest in view of the possibility that its characteristics may be related to these diseases. In addition a precise knowledge of the velocity profiles in the arterial tree is of importance for the development of flow measurement techniques, such as the ultrasonic method, allowing non-traumatic in ciao investigations. Most studies, related to the fluid mechanics in the arterial tree, are partial studies concerning the aortic valve, the ascending aorta, the aortic arch, the descending aorta or the iliac bifurcation. The flow velocity pattern, distal to different valve prostheses mounted in straight rigid tubes, has been considered in steady conditions using ultramicroscope and laser anemometry by Schramm et at. (1980) and Yoganathan rt al. (1979, 1980a,b,c). The pulsatility of the flow has been taken into account by Wieting rf al. (1972) and Alchas et al. (1980). In their reports, Chandran and Yearwood (1981), Chandran et al. (1979. 1983) and Yearwood and Chandran (1982) introduced the role played by flow pulsatility and curvature effects in a rigid aortic arch model. Several in Receircd 28 June 19%; in rerisedform
28 March
METHODS The general schema of the test bench is given in Fig. I. Since the details of the cardiovascular simulator
1955. 703
R. RIEL’. A. FRIGGI and R. PELISSIER
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Ge&;era:orsynthetlser Fig. I. Schematic of the hydrodynamic simulator have been presented elsewhere (Pelissier ef nl., 1983) we only give its main features. The motion of the fluid contained in a primary circuit (water) is obtained with an hydrodynamic generator {lssartier et at., 1978). This fluid activates an atrioventricular prosthesis that imposes the motion of another fluid (33 y/bglycerol-water mixture with starch particles) through an aortic valve in a secondary circuit including the arterial tree. This one is fixed horizontally in a piexiglas box filled up with water. In order to improve the capacitive etfects we have incorporated a compliance (Windkessei) at the output of the model. The choice of this position in the circuit was imposed by technical reasons. Resistive effects were modelled by an adjustable resistance composed of a parallel series of elastic tubes of small diameter more or less crushed between two Plexiglas plates. After passing the resistance and the compliance, the fluid flowed back to the left atrium. This simulator can reproduce accurately the in cico signals of pressure and flow rate in the ventricle and in the aorta (Farahifar et al., 1983). The arterial tree model, smooth, transparent and viscoelastic was made up of two elements stuck together. One part was produced from an in situ casting of an aortic arch and the other was made from a simplified glass model of the thoracic aorta in&ding the iliac bifurcation. The initial cast served toconstruct a glass matrix. The elastic vessel was formed from the two glass matrices using the methodology developed in the Aerodynamisches lnstitut of Aachen. The poiyurethane was thinly coated on the inner surface of the glass matrix and the cast was rotated at low speed. After setting for 24 h the same operation was repeated until the obtained wall thickness made the model sufficiently strong (here 0.7 mm). This wall thickness is
lesser than that of human aorta but a more important one would give a model with greater rigidity. The interest of such a model is its compliance which is much more realistic than many previous models. The elastic properties and the input impedance graph presented below, compared to the human results confirm our choice. The dimensions (in mm) of the model, the cross section measurements and some definitions used in the text such as inner wail, outer wall, apex. vertex were given in Fig. 2. The prosthetic aortic valve was a Starr-Edwards stellite (model 2400) ball valve with a tissue annulus diameter of 24 mm. Flow rates were recorded. using an electromagnetic Howmeter (Statham SP 2203) and three rxtracorporeai probes, in three sites, viz: at the output of the three major arterial branching at the mid-arch, at the output of one iliac artery and at the input of the ascending aorta. A polyethylene catheter was inserted into the model of the arterial tree and the pressure was measured with a Bentley Trantec, model 800, pressure transducer behind the aorric valve, in the descending aorta and in an iliac artery. Diameter variations were measured as the transit time of ultrasonic acoustic pulses travelling at a velocity of 1.58 mm@’ between pairs of 5 MHz piezo-electric crystals glued to the arterial wall (Friggi 1981). Two-dimensional velocity and Bodard, measurements were performed using a pulsed Doppler ultrasonic velocimeter originally developed by Peronne+.t rt al. (1970).This system sends out bursts of 8 MHz ultra sounds at regular intervals (repetition frequency 15 KHz). During each period between two bursts, it is possible by using an electronic gate to detect the back scattered sound and the Doppler frequency from selected points along the path of the
Fig. 2. Dimensions
of the model (in mm) and positions of the cross-sections made.
where the measurements were
reverse flow. At the output of the bifurcation branches, sound beam. The probe used was a single piezo-electric the maximum of the negative flow rate is 40 O0 of the crystal (diameter 2 mm, thickness 0. I mm), transmitter maximum positive value. In several cross-sections, and receiver, mounted on a block of Plexiglas which imposed the beam direction. The order of magnitude we have represented pressure and diameter waveforms and the corresponding pressure-diameter diagof the sample volume is 1 mm. Measurements were performed in fourteen crossram. Due to the combined effect of elasticity and sections of the model, in a horizontal plane including viscosity, the pressurediameter diagrams show a the mean line of the model (horizontal measurements) hysteresis loop. An important parameter describing and in an orthogonal plane (vertical measurements). In vessel wall properties in the distensibility D. defined by each cross-section, velocity components C’, and k’L D = (l/A)(dA/dPrm) were measured at discrete points separated by a 2 mm gap in the aortic arch and a 1 mm one in the iliac where A is the mean area of the aortic section, d A the bifurcation. I-low rate. pressure, diameter changes and variation of the area and dPtm the variation of the velocity waveforms were recorded both on a Tektronix transmural pressure during a pulsatile cycle. In our memory oscillograph and on a desktop computer. The model, the values of D computed at the level of ‘aortic experimental procedure, including the choice of the arch’ and ‘thoracic aorta’ vary from 4 to gate depth, sampling, averaging and digital storage, 5.10-3 m’kN_‘. As discussed hereafter, these values was controlled by a computer. The construction of the are probably lower than the in ciro ones. In order to velocity pattern (longitudinal component k’_ radial determine the rheological behaviour of our model, we component 1;. and velocity vector V) was achieved at have represented in Fig. 4a the input impedance graph the end of the experiment with the same microcoma few centimeters behind the valve. The mean puter from the two velocity informations b’, and C,. term (zero Hz, peripheral resistance) is about Sign convzntionr for C:, and V, are defined in Fig. 2. 4C00 dyn cm-’ and the characteristic impedance The bifurcation is symmetric with a branch-to-trunck is about 400. Characteristic impedance is calculated area ratio of I. I. The angle formed by the branches is from the pressure-diameter relation and the 70-. The vertex and the apex of the bifurcation are Moens-Korteweg equation (Merillon et ul., 1978). smooth and blunt. The branches are straight and Although these values are higher than the human ones. untapered and their lengths are equal (100 mm). The the shape of the impedance spectra of the simulated values of the bifurcation angle and of the branch-tosignals agree qualitatively with the corresponding one trunck area ratio are within the range of those of input impedance in man reported by Murgo er al. observed in riro (Walburn and Stein, 1981). (1981) (Fig. 4b). These important values of impedance modulus are certainly explained by the lack of elasticity of our model. RESULTS Velocity profiles are shown in Figs 5-10. Firstly, we Flow rate conditions in three sites along the model consider the flow in the ascending aorta, the aortic arch are shown in Fig. 3. The period duration was 1 s. The and the descending aorta. Horizontal and vertical maximum flow rate in the three derivations of the midmeasurements of the longitudinal velocity component arch was 1 I.min- ’ , about l/6 of the total flow rate. The V, are shown in Fig. 5. We first report horizontal measurements. In early systole (1 = f2) velocity profiles flow rate waveforms did obviously show an important
704
R. RIEU, A. FRIGGIand R.
PELISSIER
Top wall
Inner wal(
Pressure
(I W
(TW
)
(mmHg)
s
Fig. 3. Sites of pressure and diameter variation records, pressure diameter loops and flow rate conditions.
Frequency
Fig.
4.
(Hz1
(a) Input impedance graph of the model. (b) Graph ofmodulus of the input impedance of the human ascending aorta (Reported by Murgo et al.. 1981).
1
Velocity distrtbution
707
Fig. 5. Spatial and temporal variations of the longitudinal velocity component in the aortic arch (horizontal profile [H.P.] and vertical profiles [V.P.]). I.W.: inner wall: O.W.: outer wall; T.W.: top wall; B.W.: bottom wall. without important asymmetries. As systole progresses (1 = rs) velocities increase. In the ball wake (section A), the velocity profile shows an important minimum and two maxima, the maximum near the inner wall (I.W.) of the arch being higher than the one near the outer wall (O.W.). In sections A, B, C, D, E, the maximum velocity is located near the inner wall. In early systolic deceleration (t = t9) the shape of the are regular
velocity profile in section A is symmetric with three maxima and two minima. in sections B and C, we always notice a maximum velocity near the inner wall. In section G, the profile is blunt as during systolic acceleration. For I = I!~, in section A. the negative velocities in the central part of the tube. become very important, the flow rate upstream from the valve being small but positive. From a general point of view the
R. RIEU,A. FRIGGIand R. PELISSIER
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w
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@ @
E
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Fig. 6.
Spatial and temporal
variations
@ F
@‘ae G
of the radial velocity component in the aortic arch (horizontal and vertical profiles [V.P.] ).
profiles [H.P.]
velocity decreases along the inner wall. According to the flow rate upstream from the valve, for t = r14, during the diastole, velocity in section A is negative which leads to think to a regurgitation flow. On the contrary, velocity in section B is always positive, the velocity profile being asymmetric with a maximum value near the inner wall. In sections C. D and E the
velocity decreases, observed near the inner wall for c = c,~, leads to a reverse flow in these sections. For r = tzO,the flow rate is negative in the cross-sections B, C, D, E, F, G, but in section A the mean velocity is very low. In section G, the velocity profile shows a minimum near the tube axis. Such a shape was induced, as it will be seen later, by the bifurcation effect. During
Velocity distribution
BW
/
Fig. 7. Spatial and temporal
variations
of the velocity vector in theaorticarch vertical profiles[V.P.]).
the diastole, between sections I?, C, D, E, F, the maximum velocity vacillates between the inner and the outer wall. The vertical measurements show the following characteristics. In early systole (I = tL)velocity profiles are symmetric with an exception in section B where the velocity profile shows an asymmetric M-form with a maximum near the top wall (T.W.). As systole prog-
(horizontal profiles [H.P.] and
resses, this asymmetric M-shape is amplified. In the bend, contrary to the horizontal profiles, the vertical profiles remain approximately symmetrical and very much blunted. Very blunted velocity profiles are also observed in section G during systole while M-form appears in the bend and in the descending aorta during systolic deceleration and diastole. Radial velocity patterns are shown in Fig. 6. The
710
R. RIEL A. FRIGGIand R. PELISSIER
Fig. 8. Spatial and temporal vatialions of the longitudinal
velocity component (horizontal profiles [H.P.] and vertical profiles [V.P.]).
of these radial velocities decreases along the arterial tree and they are mainiy noticeable in section B with, for horizontal measurements, a maximum amplitude of about 10 cm s- ’ during systolic deceleration. In section B, for I = f.,. tg and 112,the radial velocities
amplitude
in the iliac bifurcation
are positive near the inner and the top walls. They are negative near the outer and the bottom (B.W.) walls. According to the existence of a maximum of the longitudinal velocity component near the inner and the top walls, negative radial velocities disappear for
Velocity distribution
= 3r =
= :
ji
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/r-?l
F’3’,il?S
@ J
Fig. 9. Spatial and temporal
variations profiles
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j-7 ,,t-----
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of the radial velocity component in the iliac bifurcation [H.P.] and vertical profiles [V.P.]).
r = r14, rrs, tr6. In section C,in the bend, thebehaviour of the radial velocity components follows the inward shift of the maximum longitudinal velocity component and the reverse flow is observed near the inner wall. These radial velocities mainly increase during the reverse flow.
-
(horizontal
Velocity vector behaviour is shown in Fig. 7. The interest of such a pattern is the illustration of the vorticity creation. Thus, between I = f6, lp and r, 2, we can observe (section A), in the ball wake, the formation and the development of two contrarotating vortices symmetrical with respect to the tube axis. Between
R. RI&U,A. FRIGGIand R.
712
PELISSIER
Horizontoi mof i!es
V.P
V.? Tw
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Fig. 10. Spatial and temporal vaiiations of the velocity vector in the iliac bifurcation (horizontal profiles [H.P.] and vertical profiles [V.P.]). I = 112. t14, [I5 and 11, it is possible to visualize the propagation of the reverse flow along the inner wall. The reverse flow is accompanied by velocity vector rotation and then by vorticitycreation. For I = rlr. the reverse flow is well pronounced in section C and less pronounced in section D. We observe a beginning of reverse Row in section E but no effect in sections F and
G. Reverse flow appears in section F for c = flJ and in section G for r = rt6. With respect to sections C, D, E and F, the reverse flow in section G is different as it is not accompanied by velocity vector rotation, i.e. vorticity creation. In the vertical display of the velocity vectors, profiles are very blunted (I = tg, f9, t,2) in section G. Between
Velocity distribution
713
files in early systole i = I, whdjr disturbances appear t = ft:. III, I,s, vorticity creation is illustrated in the for i = lp. Although lrss cltir than for the horizontal central part of the tube (sections Bt D. E. F) and near measurements. v-elocity vector rotation and then vorthe top wall lsection C). Concerning the flow in the iliac ticitycreationarzvisiblc in xctw11 J tr = r,i.~,~.r,,lin bifurcation. we only illustrate the results in one the central part of the tube and near the top and the daughter branch as both daughter branches are symbottom walls (I = I:~. I :), f 2dl.In section L no vorticity metrical. Horizontal and vertical profiles of the longicreation is revealed. tudinal velocity component are shown in Fig. 8. For horizontal measurements, in early systole (I = I,) we noted symmetrical velocity profiles in the DlSClSSlOI mother branch. In the daughter branch, the maximum This paper gives an illustration of the velocity velocity vacillates between the inner wall (section J) pattern in a model of human arterial tree. Prior to and the outer wall (section K). During the systolic discussing our results. two remarks may be fordeceleration (I = I,,), a reverse flow appears mulated. The first one concerns the elasticity of our downstream from the vertex in section J. When the model. Calculation of the distensibility of our model flow rate becomes negative (t = f,*) velocity profiles gives a mean value of 4.5 x 10-j mr kN- ‘. In canine are symmetric in the daughter branch while the thoracic aorta, Pedley (1980) reported a value of about v-elocity profile in section H shows the M-form previously described in s&ion G. For the vertical pat2x IO-‘m’kN_‘.S o, we think that our model is not sufficiently distensible and this lack of elasticity reterns. an M-form of the longitudinal velocity component profile is illustrated in section J during the duces its capacitive elfects and hence the exponential systolic acceleration and the beginning of systolic decrease of the aortic pressure during the diastole. in . deceleration (t = L,, ry. t,J. From a general pomt ot addition, this lack of elasticity, in spite of the adjuncview, vertical profiles are more blunted than the tion of a windkessel in the circuit. explains the horizontal ones. During the flow rate change. negative importance of the negative flow rate in the iliac artery. vclocitics appear at first near the top and the bottom Another remark concerns the geometry of the model. wall. For c = f,9 (negative Row rate), velocity profiles The geometry of the aortic arch is probably an are parabolic in the daughter branch and Rat in the important parameter in the behaviour of the aortic mother branch. This remark can be attributed to the flow, mainly in the ascending aorta. The casting of our ditTerent values of the Womersley’s parameter in the aortic arch model was made from an 80 yr old woman’s mother branch and in the daughter branch. and the surface including the mean line of the arch, is a surface with an important curvature. However from Figure 9 gives an illustration of the radial velocity patterns. During the flow period, radial velocities in the analysis of our results. it appears that the main station K are negligible. In section J, as systole features of the flow confirm the findings of other progresses, radial velocities increase with maximum authors. In the ascending aorta. longitudinal velocity values as high as 2Ocms-‘. Radial velocities are profile is not a flat one but the maximum velocity, positive near the inner wall and negative near the outer during the major part of the flow period. is skewed wall. They are very small during the negative flow rate towards the inner and the top walls. The skewness of corresponding to parallel flow in the daughter branthe velocity profile in the ascending dog’s aorta was ches. In section I, due to the symmetry of the flow previously reported by Schultz rr uf. (1969). Seed and between the two daughter branches, radial velocities in Wood (1971). Paulsen and Hasenkam (1983) from hot vertical measurements are very small. In horizontal film anemometry and by Peronneau Ed al. (1973) from measurements we noted an approximate symmetry 01 ultrasonic Doppler pulsed anemometry. Paulsen and these velocities: positive near the inner wall and Hasenkam found the skewness of the velocity profiles 5 cm above the aortic valve, with highest velocity negative near the outer wall. Because of the vertex nearer to the posterior and left vessel wall. The effect, their magnitudes remain of the same order asymmetry of the velocity profile. a few centimeters during the reverse flow. The behaviour of the velocity above the aortic valve must probably be attributed to vector is illustrated in Fig. 9. We first consider the horizontal measurements. In section L, the velocity the curvature of the aortic arch near the valve orifice increases during the systolic acceleration (t = r.,. r9) and possible misalignment betw-een the axis of the and decreases during systolic deceleration (I = t,2. ventricle output and the axis of the ascending ao,rta. I,~). The velocity vectors remain axial, without roMeasurement of cardiac output with a pulsed tation when the transition between positive and negatultrasonic Doppler flowmeter requires a detailed knowledge of the velocity profile along the ascending ive flow rate is engaged. In sections I and J. near the vertex. as systole progresses, we denoted a counteraorta. According to previous data, it appears thraugh clockwise rotation of the velocity vector. The zone our observations that the velocity profile is not flat a few centimeters beyond the aortic valve. In this part of interested by this rotation is located near the vertex in the early systole, but concerns an increasing part of the the aortic arch, it is also important to notice the small section J when f increases. This velocity vector rotation disturbances effect on the post valvular flow field due IS accompanied by vorticity creation. to the Starr-Edwards prosthesis. Both pulsatile flow Vertical measurements show reeular velocitv,r oroand elasticity of the wall play probably an important
711
R. RIEU. A. FRIGGI and R. PELISSIER
role in the damping. Indeed. for in rirro tests in rigid and steady conditions. the velocity profiles reported by Yoganathan er al. (198Oa).42 mm beyond the valve, always show an important valve elTect. On steady Row in a rigid model of aorta with a Starr-Edwards valve, Khalighi et at. (1983) observed velocity profiles qualitatively comparable to those obtained in our model during the systolic phase ejection. The damping due to the elasticity cannot be quantitatively evaluated with regard to the rigid model used in other studies because of differences in the geometrical configurations. Nevertheless, our results which took into account the elasticity of the wall and the pttls3tility of the flow showed that velocity prohlcs were highly modified during the cycle. For horizontal measurements in the successive cross-sections of the aortic arch the maximum longitudinal velocity vacillates between the inner and the outer wall. During the systolic acceleration we observe in the curved part of the arch an inward displacement of the maximum velocity. This displacement previously described by Farthing and Peronneau (I 979) and Chandran and Yearwood (1981) is associated to a predominance of the radial pressuregradient effects on the centrifugal forces. Reverse flow is present along the inner wall of the aortic arch during the systolic deceleration and diastole. This reverse flow, previously described by Yearwood and Chandran (1982) is accompanied by a velocity vector rotation during the cycle and then by a vorticity creation along the inner wali. In the vertical transverse, the longitudinal velocity profiles in the curved part of the aortic arch remained relatively flat in all the cross-sections but no reversal flow was observed. Mean longitudinal velocities of about 30cms-‘, 4Ocms-’ and 50cms-’ are respectively obtained in the ascending aorta, in the aortic arch and in the descending aorta. Such values are in the range of in uico values. Concerning the iliac bifurcation, the most striking features of the flow are the damping of the branching effect, the displacement of the maximum velocity and the vorticity creation near the vertex. The results obtained confirm the findings of Batten and Nerem (1982) and those of two authors of the present paper (Siouffi et al., 1984) in the study of steady and pulsatile flows in bifurcations with rectangular cross-sections and various angles. Indeed it was shown that the pulsatihty of the flow is an important parameter in the cardiovascular fluid dynamics. In daughter branches of a bifurcation the reversion to an axisymmetric pattern occurs more quickly for putsatile than for steady flow. The distance from the apex for this reversion decreases when the frequency parameter increases. At the entry of the daughter branch, the skewing of the horizontal profile is located towards the inner wall for steady flow and towards the outer wall for pulsatile flow. The inward shift of the maximum velocity is an increasing function of the frequency parameter. In addition the amplitude of the secondary velocities decreases when the frequency parameter increases. For the present study, tubes
we have obtained, in section J in the entrance of the daughter branch. secondary velocities as high as 30 o,t, of the longitudinal component. However, secondary velocities are not noticeable in the other cross-sections considered. For the horizontal measurements it is important to emphasize the illustration, near the vertex, of vorticity creation by velocity vector rotation. Such illustration. not usual in the literature, put into evidence thezone of interest for blood ceil damage. In vertical transverse, Batten and Nerem (19S?) noted M-form in the inlet of the daughter branches. Such a shape obtained herein can be explained by the importance of the centrifugal effect in the central part of the tube. M-form is also observed in horizontal and vertical transverses of the mother branch during the negative flow rate. In this case this shape has to be attributed to the bifurcation effect. In conclusion it can be drawn from the present study: (I) Flow in the ascending aorta is not most influenced by the aortic valve. (2) Skewness of the velocity profile, with higher velocities towards the top and the inner wall, is observed a few centimeters beyond the valve. (3) Inward shift of the maximum velocity and reverse flow, accompanied by vorticity creation, are denoted along the inner wall of the aortic arch. (4) The velocity profiles in the descending aorta are blunted. (5) Yorticity creation can be seen downstream from the vertex of the iliac bifurcation. (6) The branching effect due to the iliac bifurcation is quickly damped by the pulsatility of the flow and the elasticity of the wall. kknowledgemenr-We are very grateful to the Aerodynamisches Institut of Aachen (Germany) for their contributing to moulding realization and to the SonderFroschungsbereich 109 for providing US with the atrioventricular prosthesis.
REFERENCES Alchas, P. G., Snyder, A. J. and Phillips, W. M. (1980) Pulsatile prosthetic valve flows: laser Doppler studies. Bicy?uid Mechunics (Edited by Schneck, D.), pp. 243265.
Plenum Press, New York. Batten, J. R. and Nerem, R. M. (1982) Model study of flow in curved and planar arterial bifurcations. Curdiowsc. Res. 16, 178-186. Bergel, D. M. (1972) Cardiocuscular Fluid DYM!?&zs, Vol. 2. Academic Press, London, New York. Brah, R. and Bellhouse, 8. J. (1973) Flow in branching vessels.Cardiovaw. Rex 7, 593400. Caro, C. G., Fitz-Gerald. J. M. and Shroter, R. C. (1971) Atheroma and arterial wall shear observation: correlation and proposal ofa shear dependent mass transfer meehankm for atherogenesis. Pwc. R. SOC.,Land. B117,109-S59. Chandran, K. 9. and Yearwood, T. L. (1981) Experimental study of physiological pukatile flow in a curved tube. 1. Fluid Mech. 11, 59-85. Chandran, K. 9.. CabelI, G. N., Khalighi, 9. and Chen. C. J.
Ycloclty distribution t i 983 t Laser anemomstry measurements of pulsatile Bow post aortic valve prostheses. J. Biomechanics 16, bhS-873 Chandran. K. B.. Yearwood. T. L. and Wieting. D. W. (1979) An experimental study of pulsatile fiow in a curved tube. J. Biantt~-honics 12, 793-SOS. Fa!sctti. H. L., Kiser. K. M.. Francis. G. P. and Belmore, E. R. / 1972)Sequential velocity development in the ascending and descending aorta of the dog. Circularion Rex 31, 3:s-33g FarthIng. S. and Peronneau. P. (1979) Flow in the thoracic aorta. Curdiotasc. Rrs. 13, 607620. Farahifdr, D., Cassot. F.. Bodard. H. and Pelissier, R. (1983) A cardiovascular pulse duplicator for the in cirroasessment of artificial heart valves under physiological conditions. J. Eur. Sac. orrij: Organs. Life Support SystrrW 1, Suppl. 1, 33:-335. Frtpgi. A. and Bodard. H. (1981) Effects ofclonidine on aortic p&sure-diameter and elastic stiffness-stress relations in anaesthetized rabbtts. C/in. Sci. 61. 273279. F-ry. D. L. (1968) Acute vascular endothelial changes associated wtth increased blood velocity gradients. Circulation Res. 22, 165-195. Khahghi. B.. Chandran. K. B. and Chen. C. J. (1983) Steady flow development past valve prostheses in model human aorta I-Centrally occluding valves. J. Biomechanirs 16, 1003-101 I. Issartier, P.. Siouffi. M. and Pelissier. R. (1978) Simulation of blood flow by an hydrodynamic generator. Med. Pray. Technol.. 6, 39-40. Ling, S. C. and Atabek. H. B. (1972) A non linear analysis of pulratile flow in arteries. J. Fluid Mech. 55, 493. Ling, S. C.. Atabck. H. B., Fry, D. L.. Patek D. L. and Janicki, J. S. (196s) Applicatton of heated-film velocity and shear probes to haemodynamic studies. Circulution Res. 23, 789 %!I. Merillon. J. P., Motte. G., Fruchaud. J., Masquet, C. and Gourpon. R. (1978) Evaluation of the elasticity and characteristic impedance of the ascending aorta in man. Curdiocusc. Rex 12. 4Oi 406. ?vfuurgo. J. P., Westerhof. N., Giolma, J. P. and Altobelli. S. A. (1981) Effects of exercise on aortic input impedance and pressure wave forms m normal human. Circularion Rex 48. 334-343. Olson. D. E. (1971) Flutd mechanics relevant to respiration How within curved or elliptal tubes and bifurcating systems. PhD thesis. Imperial College, London. Pdtcl. D. J.. Janicki, J S.. Vaishnav. R. N. and Young, J. T. (1973) Dynamic antsotropic viscoelastic properties of the aorta in hving dogs. Circulation Res. 32, 93. P&srn. P. K. and Hasenkam. J. P. (1983jThreedimensional visualization of velocity protiles in the ascending aorta in dogs. measured wnh a hot-film anemometer. J. Biamechanics 16, 201-210. Pcdley. T. J. (1980) 7hr Fluid Mechanics a/ Lorye Hood C’essels. Cambridge University Press. Pehssier. R., Cassot, F., Farahifar. D.. Bialonski. W., Issartier, P.. Bodard, H. and Friggi. A. (1983) Simulateur cardiavasc. Innor. Technol. &al. ,Cfed., 4, 33-45. Peronneau, P.. Hinglais. J., Xhaard, M., Delouche, P. and Philippo. J. (1973) Cardiovascular applications of ultrasound. Procredtngs a/ on International Symposium held at Janssen Phormoceurica, Beerse, Belgium. (Edited by Renemdn. R. S.). North Holland, Amsterdam. Peronneau, P.. Leger, F., Htnglais, 1.. Pellet, M. and Schwartz.
715
P. Y. (1970) Velocimetre sanguin a etfet Doppler B emission ultrasonore pub& Ondr ilect. 50. 369-389. Rieu. R.. Issartier. P.. Prlissier. R.. Mouttet, J. P.. Palomba. P. and Peronn&u, P. (1979) Experimental investtgation of the flow in a vascular bifurcation. Euramech 11% CardiaL-ascukv and Puimonar~ Dynamics. Editions Inserm, Paris. Schramm. D.. Baidauf, W. and hleisner, H. (1980) Flow pattern and veloctty field distal to human aortic and artificial heart valves as me;lnured simultaneously by ultramicroscope antmometry in cylindrical glass tubes. 7Ioroc. Curdrolasc. Sury. 28, 133-140. Schroter. R. C. and Sudlow. M. F. (1969) Flow patterns in models of the human bronchial airways. Resp. Phys. 7, 34l-3%. Schultr D. L.. Tunstall-pedoe, D. S., Lee, G. de J., Gunning. A. J. and Belihousc. B. J. (1969) Velocclty distribution and transttion in the arterial system. Circuiarory and Respirator) ,Vfass Transport. (Edtted by Wolstcnholme. G. E. W. and Knight. J.) A CUBA Foundation Symposium, London. Seed, W. A. and Wood, N. B. (1971) Velocity patterns in the aorta. Cardiorasc. Res. 5, 319330. Siouffi. M.. Pelissier. R.. Farahifar. D. and Rieu. R. (1984)The effect of unste;ldiness on the flow through stenoses and bifurcations. i. Biomrchanics 17. 299-3 15. Talukder. N. (1975) An investigation on the flow charactcristics in arterial branchings. Truns. Am. Sot. Mech. Engr. 15, 4. Taylor. L. A. and Gerrard. J. H. (1977) Pressure radius relationships for elastic tubes and their applications to arteries. Med. bid. Engny Compur. 15, 11. Tomm, D. (1978) Stromuug ud Gerausch in verengten und verzweigten gefaben. Diplom Ingenieur. Aerodynamisch Institut (Aachen). Walburn. J. W. and Stein, P. D. (1981) Velocity profiles in symmetrically branched tubes simulating the aortic bifurcation. J. Biomechonics. 14, 601411. Wetterer. E.. Bauer. R. D. and Busse, R. (1977) Arterial dynamics. CardioLascular and Pulmonary Dynamics, Euromah 92, Publication INSERM. Wieting, D. N., Kennedy, J. H. and Hwang, N. H. C. (1972) Testing of heart valve replacements. Adv. Cordial. 7, 2-11. Yearwood, T. L. and Chandran. K. B. (1982) Physiological puktife flow experiments in a model of the human aortic arch. J. Eiomechanics 15, 683-704. Yoganathan, A. P., Corcoran. W. H. and Harrtson. E. C. (1979) In t-itro measurements in the vicinity of aortic prostheses. J. Biomerhanics 12, 135-l 52. Yoganathan, A. P.. Corcoran. W. H., Harrison, E. C. and Chaux, A. (1980b) In citro fluid dynamics of St Jude, Ioncscu-Shiley and Carpentier-Edwards aortic heart valve prostheses. Biojuid ?nechanics (Edited by Schneck, D. J.) pp. 295-316. Plenum Press, New- York. Yoganathan. A. P., Reamer, H. H.. Corcoran. W. H. and Harrison, E. C. (198Oc) The Bjork-Shiley aortic prosthesis: flow characteristics of the present model vs the convexoconcave model. Stand. 1. Thorac. Cardiocasc Surg. 14, l-5. Yoganathan, A. P., Reamer, H. H., Corcoran. W. H.. Harrison, E. C., Schulman, 1. A. and Parnassus. W. (198&a) The Starr-Edwards aortic ball valve: flow characteristics, thrombus formation and tissue overgrowth. Biofluid Mrrhonics Vol. 2 (Edited by Schneck. D.). Plenum Press, New York.
DISTRIBUTION ALONG AN ELASTIC OF HUMAN ARTERIAL TREE
In>tirut de hticaniqut:
des Fluidcs.
MODEL
1 Rue Honnorat. I XX73Marseille. France and
ALAIN
Unit6 Inserm 175, Cardiologie
FRIGGI and ROBERT PELISSIER ExpGmentale. lg Avenue Mozart. l?@O!J. Marseille, France
Abstract-An experimental investigation ofan elastic model ofthe human arterial tree, has ken performed ior physiological type flow by pulsed Doppler ultrasonic velocimetry. The arterial tree model, fabricated in clear poiyurrthanc, includes the aortic arch, with a Starr-Edwards ball valve mounted in the root of the aorta, the descending aorta and the iliac bifurcation. Our study showed that the velocity profile, a few centimeters beyond the valve, is skcwcd, with higher velocities towards the top and the inner wall (anatomically the posterior and left lateral wall). An inward shift of the maximum velocity and reverse flow are denoted along the inner wall of the aortic arch. The velocity profiles in the descending aorta are blunted. Downstream from the vertex ofthe iliac bifurcation. thcrc is vorticitv creation, but the branching effect is quickly damped by the pulsatility of the flow and the elasticity of the ;ail.
rit.0 attempts have been made to describe the velocity profiles in the aorta using hot-film anemometry (Ling er al., 1968; Schultz et al., 1969; Seed and Wood, 1971; Falsetti et ul., 1972; Paulsen and Hasenkam, 1983) or pulsed Doppler ultrasonic anemometry (Farthing and Peronneau, 1979). Numerous studies have been reported in the literature which investigate flow characteristics in branching channels. A large review was given by Pedley (1980). The flow velocity pattern has been considered in steady conditions by Schroter and Sudlow (1969). Olson (1971). Brech and Bellhouse (1973). Talukder (1975) and Tomm (1978). Pulsatile flow studies have been reported by Brech and Bellhouse (19731, Rieu et al. (1979). Walburn and Stein (1981), Batten and Nerem (1982) and Siouffi et a/. (1984). The rheological properties of the arterial wall are of interest in fluid dynamics. Elasticity is of importance but is neglected by most investigators in this area of research. In reality, the elastic behaviour of the wall is non-linear; this non-linearity of the wall elements was clearly proved by Ling and Atabek (1972), Pate1 et al. (1973). Taylor and Gerrard (1977), Wetterer et al. (1977) and Friggi and Bodard (1981). The purpose of the present paper was to determine, under physiological type flow conditions, the velocity profiles in several cross-sections of an elastic model of arterial tree including a Starr-Edwards valve prosthesis. Velocity measurements were performed for the longitudinal and the radial velocity components and afterwards for the velocity vectors.
INTRODUCTIOS
The description of the flow pattern in the arterial tree is of basic importance to understanding the fluid mechanics of the circulation. The very complex nature of the phenomena in the arterial tree is due to the fact that the flow is disturbed by many singularities: aortic valve, aortic arch, branches at the arch and iliac bifurcation. Valve implantations remain associated with thrombus formation and tissue overgrowth (Yoganathan et a[.. 198Oa);sites of curvature and bifurcations are known to be particularly susceptible to the development of atherosclerotic lesions (Fry, 1968; Care et al., 1971). Flow in the arterial tree has been of interest in view of the possibility that its characteristics may be related to these diseases. In addition a precise knowledge of the velocity profiles in the arterial tree is of importance for the development of flow measurement techniques, such as the ultrasonic method, allowing non-traumatic in ciao investigations. Most studies, related to the fluid mechanics in the arterial tree, are partial studies concerning the aortic valve, the ascending aorta, the aortic arch, the descending aorta or the iliac bifurcation. The flow velocity pattern, distal to different valve prostheses mounted in straight rigid tubes, has been considered in steady conditions using ultramicroscope and laser anemometry by Schramm et at. (1980) and Yoganathan rt al. (1979, 1980a,b,c). The pulsatility of the flow has been taken into account by Wieting rf al. (1972) and Alchas et al. (1980). In their reports, Chandran and Yearwood (1981), Chandran et al. (1979. 1983) and Yearwood and Chandran (1982) introduced the role played by flow pulsatility and curvature effects in a rigid aortic arch model. Several in Receircd 28 June 19%; in rerisedform
28 March
METHODS The general schema of the test bench is given in Fig. I. Since the details of the cardiovascular simulator
1955. 703
R. RIEL’. A. FRIGGI and R. PELISSIER
704
Secondiir/ c8:c"I'
.‘.
Ge&;era:orsynthetlser Fig. I. Schematic of the hydrodynamic simulator have been presented elsewhere (Pelissier ef nl., 1983) we only give its main features. The motion of the fluid contained in a primary circuit (water) is obtained with an hydrodynamic generator {lssartier et at., 1978). This fluid activates an atrioventricular prosthesis that imposes the motion of another fluid (33 y/bglycerol-water mixture with starch particles) through an aortic valve in a secondary circuit including the arterial tree. This one is fixed horizontally in a piexiglas box filled up with water. In order to improve the capacitive etfects we have incorporated a compliance (Windkessei) at the output of the model. The choice of this position in the circuit was imposed by technical reasons. Resistive effects were modelled by an adjustable resistance composed of a parallel series of elastic tubes of small diameter more or less crushed between two Plexiglas plates. After passing the resistance and the compliance, the fluid flowed back to the left atrium. This simulator can reproduce accurately the in cico signals of pressure and flow rate in the ventricle and in the aorta (Farahifar et al., 1983). The arterial tree model, smooth, transparent and viscoelastic was made up of two elements stuck together. One part was produced from an in situ casting of an aortic arch and the other was made from a simplified glass model of the thoracic aorta in&ding the iliac bifurcation. The initial cast served toconstruct a glass matrix. The elastic vessel was formed from the two glass matrices using the methodology developed in the Aerodynamisches lnstitut of Aachen. The poiyurethane was thinly coated on the inner surface of the glass matrix and the cast was rotated at low speed. After setting for 24 h the same operation was repeated until the obtained wall thickness made the model sufficiently strong (here 0.7 mm). This wall thickness is
lesser than that of human aorta but a more important one would give a model with greater rigidity. The interest of such a model is its compliance which is much more realistic than many previous models. The elastic properties and the input impedance graph presented below, compared to the human results confirm our choice. The dimensions (in mm) of the model, the cross section measurements and some definitions used in the text such as inner wail, outer wall, apex. vertex were given in Fig. 2. The prosthetic aortic valve was a Starr-Edwards stellite (model 2400) ball valve with a tissue annulus diameter of 24 mm. Flow rates were recorded. using an electromagnetic Howmeter (Statham SP 2203) and three rxtracorporeai probes, in three sites, viz: at the output of the three major arterial branching at the mid-arch, at the output of one iliac artery and at the input of the ascending aorta. A polyethylene catheter was inserted into the model of the arterial tree and the pressure was measured with a Bentley Trantec, model 800, pressure transducer behind the aorric valve, in the descending aorta and in an iliac artery. Diameter variations were measured as the transit time of ultrasonic acoustic pulses travelling at a velocity of 1.58 mm@’ between pairs of 5 MHz piezo-electric crystals glued to the arterial wall (Friggi 1981). Two-dimensional velocity and Bodard, measurements were performed using a pulsed Doppler ultrasonic velocimeter originally developed by Peronne+.t rt al. (1970).This system sends out bursts of 8 MHz ultra sounds at regular intervals (repetition frequency 15 KHz). During each period between two bursts, it is possible by using an electronic gate to detect the back scattered sound and the Doppler frequency from selected points along the path of the
Fig. 2. Dimensions
of the model (in mm) and positions of the cross-sections made.
where the measurements were
reverse flow. At the output of the bifurcation branches, sound beam. The probe used was a single piezo-electric the maximum of the negative flow rate is 40 O0 of the crystal (diameter 2 mm, thickness 0. I mm), transmitter maximum positive value. In several cross-sections, and receiver, mounted on a block of Plexiglas which imposed the beam direction. The order of magnitude we have represented pressure and diameter waveforms and the corresponding pressure-diameter diagof the sample volume is 1 mm. Measurements were performed in fourteen crossram. Due to the combined effect of elasticity and sections of the model, in a horizontal plane including viscosity, the pressurediameter diagrams show a the mean line of the model (horizontal measurements) hysteresis loop. An important parameter describing and in an orthogonal plane (vertical measurements). In vessel wall properties in the distensibility D. defined by each cross-section, velocity components C’, and k’L D = (l/A)(dA/dPrm) were measured at discrete points separated by a 2 mm gap in the aortic arch and a 1 mm one in the iliac where A is the mean area of the aortic section, d A the bifurcation. I-low rate. pressure, diameter changes and variation of the area and dPtm the variation of the velocity waveforms were recorded both on a Tektronix transmural pressure during a pulsatile cycle. In our memory oscillograph and on a desktop computer. The model, the values of D computed at the level of ‘aortic experimental procedure, including the choice of the arch’ and ‘thoracic aorta’ vary from 4 to gate depth, sampling, averaging and digital storage, 5.10-3 m’kN_‘. As discussed hereafter, these values was controlled by a computer. The construction of the are probably lower than the in ciro ones. In order to velocity pattern (longitudinal component k’_ radial determine the rheological behaviour of our model, we component 1;. and velocity vector V) was achieved at have represented in Fig. 4a the input impedance graph the end of the experiment with the same microcoma few centimeters behind the valve. The mean puter from the two velocity informations b’, and C,. term (zero Hz, peripheral resistance) is about Sign convzntionr for C:, and V, are defined in Fig. 2. 4C00 dyn cm-’ and the characteristic impedance The bifurcation is symmetric with a branch-to-trunck is about 400. Characteristic impedance is calculated area ratio of I. I. The angle formed by the branches is from the pressure-diameter relation and the 70-. The vertex and the apex of the bifurcation are Moens-Korteweg equation (Merillon et ul., 1978). smooth and blunt. The branches are straight and Although these values are higher than the human ones. untapered and their lengths are equal (100 mm). The the shape of the impedance spectra of the simulated values of the bifurcation angle and of the branch-tosignals agree qualitatively with the corresponding one trunck area ratio are within the range of those of input impedance in man reported by Murgo er al. observed in riro (Walburn and Stein, 1981). (1981) (Fig. 4b). These important values of impedance modulus are certainly explained by the lack of elasticity of our model. RESULTS Velocity profiles are shown in Figs 5-10. Firstly, we Flow rate conditions in three sites along the model consider the flow in the ascending aorta, the aortic arch are shown in Fig. 3. The period duration was 1 s. The and the descending aorta. Horizontal and vertical maximum flow rate in the three derivations of the midmeasurements of the longitudinal velocity component arch was 1 I.min- ’ , about l/6 of the total flow rate. The V, are shown in Fig. 5. We first report horizontal measurements. In early systole (1 = f2) velocity profiles flow rate waveforms did obviously show an important
704
R. RIEU, A. FRIGGIand R.
PELISSIER
Top wall
Inner wal(
Pressure
(I W
(TW
)
(mmHg)
s
Fig. 3. Sites of pressure and diameter variation records, pressure diameter loops and flow rate conditions.
Frequency
Fig.
4.
(Hz1
(a) Input impedance graph of the model. (b) Graph ofmodulus of the input impedance of the human ascending aorta (Reported by Murgo et al.. 1981).
1
Velocity distrtbution
707
Fig. 5. Spatial and temporal variations of the longitudinal velocity component in the aortic arch (horizontal profile [H.P.] and vertical profiles [V.P.]). I.W.: inner wall: O.W.: outer wall; T.W.: top wall; B.W.: bottom wall. without important asymmetries. As systole progresses (1 = rs) velocities increase. In the ball wake (section A), the velocity profile shows an important minimum and two maxima, the maximum near the inner wall (I.W.) of the arch being higher than the one near the outer wall (O.W.). In sections A, B, C, D, E, the maximum velocity is located near the inner wall. In early systolic deceleration (t = t9) the shape of the are regular
velocity profile in section A is symmetric with three maxima and two minima. in sections B and C, we always notice a maximum velocity near the inner wall. In section G, the profile is blunt as during systolic acceleration. For I = I!~, in section A. the negative velocities in the central part of the tube. become very important, the flow rate upstream from the valve being small but positive. From a general point of view the
R. RIEU,A. FRIGGIand R. PELISSIER
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w
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Fig. 6.
Spatial and temporal
variations
@ F
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of the radial velocity component in the aortic arch (horizontal and vertical profiles [V.P.] ).
profiles [H.P.]
velocity decreases along the inner wall. According to the flow rate upstream from the valve, for t = r14, during the diastole, velocity in section A is negative which leads to think to a regurgitation flow. On the contrary, velocity in section B is always positive, the velocity profile being asymmetric with a maximum value near the inner wall. In sections C. D and E the
velocity decreases, observed near the inner wall for c = c,~, leads to a reverse flow in these sections. For r = tzO,the flow rate is negative in the cross-sections B, C, D, E, F, G, but in section A the mean velocity is very low. In section G, the velocity profile shows a minimum near the tube axis. Such a shape was induced, as it will be seen later, by the bifurcation effect. During
Velocity distribution
BW
/
Fig. 7. Spatial and temporal
variations
of the velocity vector in theaorticarch vertical profiles[V.P.]).
the diastole, between sections I?, C, D, E, F, the maximum velocity vacillates between the inner and the outer wall. The vertical measurements show the following characteristics. In early systole (I = tL)velocity profiles are symmetric with an exception in section B where the velocity profile shows an asymmetric M-form with a maximum near the top wall (T.W.). As systole prog-
(horizontal profiles [H.P.] and
resses, this asymmetric M-shape is amplified. In the bend, contrary to the horizontal profiles, the vertical profiles remain approximately symmetrical and very much blunted. Very blunted velocity profiles are also observed in section G during systole while M-form appears in the bend and in the descending aorta during systolic deceleration and diastole. Radial velocity patterns are shown in Fig. 6. The
710
R. RIEL A. FRIGGIand R. PELISSIER
Fig. 8. Spatial and temporal vatialions of the longitudinal
velocity component (horizontal profiles [H.P.] and vertical profiles [V.P.]).
of these radial velocities decreases along the arterial tree and they are mainiy noticeable in section B with, for horizontal measurements, a maximum amplitude of about 10 cm s- ’ during systolic deceleration. In section B, for I = f.,. tg and 112,the radial velocities
amplitude
in the iliac bifurcation
are positive near the inner and the top walls. They are negative near the outer and the bottom (B.W.) walls. According to the existence of a maximum of the longitudinal velocity component near the inner and the top walls, negative radial velocities disappear for
Velocity distribution
= 3r =
= :
ji
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Fig. 9. Spatial and temporal
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of the radial velocity component in the iliac bifurcation [H.P.] and vertical profiles [V.P.]).
r = r14, rrs, tr6. In section C,in the bend, thebehaviour of the radial velocity components follows the inward shift of the maximum longitudinal velocity component and the reverse flow is observed near the inner wall. These radial velocities mainly increase during the reverse flow.
-
(horizontal
Velocity vector behaviour is shown in Fig. 7. The interest of such a pattern is the illustration of the vorticity creation. Thus, between I = f6, lp and r, 2, we can observe (section A), in the ball wake, the formation and the development of two contrarotating vortices symmetrical with respect to the tube axis. Between
R. RI&U,A. FRIGGIand R.
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Fig. 10. Spatial and temporal vaiiations of the velocity vector in the iliac bifurcation (horizontal profiles [H.P.] and vertical profiles [V.P.]). I = 112. t14, [I5 and 11, it is possible to visualize the propagation of the reverse flow along the inner wall. The reverse flow is accompanied by velocity vector rotation and then by vorticitycreation. For I = rlr. the reverse flow is well pronounced in section C and less pronounced in section D. We observe a beginning of reverse Row in section E but no effect in sections F and
G. Reverse flow appears in section F for c = flJ and in section G for r = rt6. With respect to sections C, D, E and F, the reverse flow in section G is different as it is not accompanied by velocity vector rotation, i.e. vorticity creation. In the vertical display of the velocity vectors, profiles are very blunted (I = tg, f9, t,2) in section G. Between
Velocity distribution
713
files in early systole i = I, whdjr disturbances appear t = ft:. III, I,s, vorticity creation is illustrated in the for i = lp. Although lrss cltir than for the horizontal central part of the tube (sections Bt D. E. F) and near measurements. v-elocity vector rotation and then vorthe top wall lsection C). Concerning the flow in the iliac ticitycreationarzvisiblc in xctw11 J tr = r,i.~,~.r,,lin bifurcation. we only illustrate the results in one the central part of the tube and near the top and the daughter branch as both daughter branches are symbottom walls (I = I:~. I :), f 2dl.In section L no vorticity metrical. Horizontal and vertical profiles of the longicreation is revealed. tudinal velocity component are shown in Fig. 8. For horizontal measurements, in early systole (I = I,) we noted symmetrical velocity profiles in the DlSClSSlOI mother branch. In the daughter branch, the maximum This paper gives an illustration of the velocity velocity vacillates between the inner wall (section J) pattern in a model of human arterial tree. Prior to and the outer wall (section K). During the systolic discussing our results. two remarks may be fordeceleration (I = I,,), a reverse flow appears mulated. The first one concerns the elasticity of our downstream from the vertex in section J. When the model. Calculation of the distensibility of our model flow rate becomes negative (t = f,*) velocity profiles gives a mean value of 4.5 x 10-j mr kN- ‘. In canine are symmetric in the daughter branch while the thoracic aorta, Pedley (1980) reported a value of about v-elocity profile in section H shows the M-form previously described in s&ion G. For the vertical pat2x IO-‘m’kN_‘.S o, we think that our model is not sufficiently distensible and this lack of elasticity reterns. an M-form of the longitudinal velocity component profile is illustrated in section J during the duces its capacitive elfects and hence the exponential systolic acceleration and the beginning of systolic decrease of the aortic pressure during the diastole. in . deceleration (t = L,, ry. t,J. From a general pomt ot addition, this lack of elasticity, in spite of the adjuncview, vertical profiles are more blunted than the tion of a windkessel in the circuit. explains the horizontal ones. During the flow rate change. negative importance of the negative flow rate in the iliac artery. vclocitics appear at first near the top and the bottom Another remark concerns the geometry of the model. wall. For c = f,9 (negative Row rate), velocity profiles The geometry of the aortic arch is probably an are parabolic in the daughter branch and Rat in the important parameter in the behaviour of the aortic mother branch. This remark can be attributed to the flow, mainly in the ascending aorta. The casting of our ditTerent values of the Womersley’s parameter in the aortic arch model was made from an 80 yr old woman’s mother branch and in the daughter branch. and the surface including the mean line of the arch, is a surface with an important curvature. However from Figure 9 gives an illustration of the radial velocity patterns. During the flow period, radial velocities in the analysis of our results. it appears that the main station K are negligible. In section J, as systole features of the flow confirm the findings of other progresses, radial velocities increase with maximum authors. In the ascending aorta. longitudinal velocity values as high as 2Ocms-‘. Radial velocities are profile is not a flat one but the maximum velocity, positive near the inner wall and negative near the outer during the major part of the flow period. is skewed wall. They are very small during the negative flow rate towards the inner and the top walls. The skewness of corresponding to parallel flow in the daughter branthe velocity profile in the ascending dog’s aorta was ches. In section I, due to the symmetry of the flow previously reported by Schultz rr uf. (1969). Seed and between the two daughter branches, radial velocities in Wood (1971). Paulsen and Hasenkam (1983) from hot vertical measurements are very small. In horizontal film anemometry and by Peronneau Ed al. (1973) from measurements we noted an approximate symmetry 01 ultrasonic Doppler pulsed anemometry. Paulsen and these velocities: positive near the inner wall and Hasenkam found the skewness of the velocity profiles 5 cm above the aortic valve, with highest velocity negative near the outer wall. Because of the vertex nearer to the posterior and left vessel wall. The effect, their magnitudes remain of the same order asymmetry of the velocity profile. a few centimeters during the reverse flow. The behaviour of the velocity above the aortic valve must probably be attributed to vector is illustrated in Fig. 9. We first consider the horizontal measurements. In section L, the velocity the curvature of the aortic arch near the valve orifice increases during the systolic acceleration (t = r.,. r9) and possible misalignment betw-een the axis of the and decreases during systolic deceleration (I = t,2. ventricle output and the axis of the ascending ao,rta. I,~). The velocity vectors remain axial, without roMeasurement of cardiac output with a pulsed tation when the transition between positive and negatultrasonic Doppler flowmeter requires a detailed knowledge of the velocity profile along the ascending ive flow rate is engaged. In sections I and J. near the vertex. as systole progresses, we denoted a counteraorta. According to previous data, it appears thraugh clockwise rotation of the velocity vector. The zone our observations that the velocity profile is not flat a few centimeters beyond the aortic valve. In this part of interested by this rotation is located near the vertex in the early systole, but concerns an increasing part of the the aortic arch, it is also important to notice the small section J when f increases. This velocity vector rotation disturbances effect on the post valvular flow field due IS accompanied by vorticity creation. to the Starr-Edwards prosthesis. Both pulsatile flow Vertical measurements show reeular velocitv,r oroand elasticity of the wall play probably an important
711
R. RIEU. A. FRIGGI and R. PELISSIER
role in the damping. Indeed. for in rirro tests in rigid and steady conditions. the velocity profiles reported by Yoganathan er al. (198Oa).42 mm beyond the valve, always show an important valve elTect. On steady Row in a rigid model of aorta with a Starr-Edwards valve, Khalighi et at. (1983) observed velocity profiles qualitatively comparable to those obtained in our model during the systolic phase ejection. The damping due to the elasticity cannot be quantitatively evaluated with regard to the rigid model used in other studies because of differences in the geometrical configurations. Nevertheless, our results which took into account the elasticity of the wall and the pttls3tility of the flow showed that velocity prohlcs were highly modified during the cycle. For horizontal measurements in the successive cross-sections of the aortic arch the maximum longitudinal velocity vacillates between the inner and the outer wall. During the systolic acceleration we observe in the curved part of the arch an inward displacement of the maximum velocity. This displacement previously described by Farthing and Peronneau (I 979) and Chandran and Yearwood (1981) is associated to a predominance of the radial pressuregradient effects on the centrifugal forces. Reverse flow is present along the inner wall of the aortic arch during the systolic deceleration and diastole. This reverse flow, previously described by Yearwood and Chandran (1982) is accompanied by a velocity vector rotation during the cycle and then by a vorticity creation along the inner wali. In the vertical transverse, the longitudinal velocity profiles in the curved part of the aortic arch remained relatively flat in all the cross-sections but no reversal flow was observed. Mean longitudinal velocities of about 30cms-‘, 4Ocms-’ and 50cms-’ are respectively obtained in the ascending aorta, in the aortic arch and in the descending aorta. Such values are in the range of in uico values. Concerning the iliac bifurcation, the most striking features of the flow are the damping of the branching effect, the displacement of the maximum velocity and the vorticity creation near the vertex. The results obtained confirm the findings of Batten and Nerem (1982) and those of two authors of the present paper (Siouffi et al., 1984) in the study of steady and pulsatile flows in bifurcations with rectangular cross-sections and various angles. Indeed it was shown that the pulsatihty of the flow is an important parameter in the cardiovascular fluid dynamics. In daughter branches of a bifurcation the reversion to an axisymmetric pattern occurs more quickly for putsatile than for steady flow. The distance from the apex for this reversion decreases when the frequency parameter increases. At the entry of the daughter branch, the skewing of the horizontal profile is located towards the inner wall for steady flow and towards the outer wall for pulsatile flow. The inward shift of the maximum velocity is an increasing function of the frequency parameter. In addition the amplitude of the secondary velocities decreases when the frequency parameter increases. For the present study, tubes
we have obtained, in section J in the entrance of the daughter branch. secondary velocities as high as 30 o,t, of the longitudinal component. However, secondary velocities are not noticeable in the other cross-sections considered. For the horizontal measurements it is important to emphasize the illustration, near the vertex, of vorticity creation by velocity vector rotation. Such illustration. not usual in the literature, put into evidence thezone of interest for blood ceil damage. In vertical transverse, Batten and Nerem (19S?) noted M-form in the inlet of the daughter branches. Such a shape obtained herein can be explained by the importance of the centrifugal effect in the central part of the tube. M-form is also observed in horizontal and vertical transverses of the mother branch during the negative flow rate. In this case this shape has to be attributed to the bifurcation effect. In conclusion it can be drawn from the present study: (I) Flow in the ascending aorta is not most influenced by the aortic valve. (2) Skewness of the velocity profile, with higher velocities towards the top and the inner wall, is observed a few centimeters beyond the valve. (3) Inward shift of the maximum velocity and reverse flow, accompanied by vorticity creation, are denoted along the inner wall of the aortic arch. (4) The velocity profiles in the descending aorta are blunted. (5) Yorticity creation can be seen downstream from the vertex of the iliac bifurcation. (6) The branching effect due to the iliac bifurcation is quickly damped by the pulsatility of the flow and the elasticity of the wall. kknowledgemenr-We are very grateful to the Aerodynamisches Institut of Aachen (Germany) for their contributing to moulding realization and to the SonderFroschungsbereich 109 for providing US with the atrioventricular prosthesis.
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